Barycenter technique and the Riemann mapping problem of Escobar

We solve the remaining cases of the Riemann mapping problem of Escobar. Indeed, performing a suitable scheme of the barycenter technique of Bahri-Coron via the Chen’s bubbles, we solve the cases left open after the work of Chen. Thus, combining our work with the ones of Almaraz, Chen, Escobar and Marques we have that every compact Riemannian manifold with boundary of dimension greater or equal than 3 carries a conformal scalar flat metric with constant mean curvature.